Theorem coe1tmmul2 22595

Description: Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)

Hypotheses

Ref	Expression
coe1tm.z	⊢ 0 = (0g‘𝑅)
coe1tm.k	⊢ 𝐾 = (Base‘𝑅)
coe1tm.p	⊢ 𝑃 = (Poly1‘𝑅)
coe1tm.x	⊢ 𝑋 = (var1‘𝑅)
coe1tm.m	⊢ · = ( ·𝑠 ‘𝑃)
coe1tm.n	⊢ 𝑁 = (mulGrp‘𝑃)
coe1tm.e	⊢ ↑ = (.g‘𝑁)
coe1tmmul.b	⊢ 𝐵 = (Base‘𝑃)
coe1tmmul.t	⊢ ∙ = (.r‘𝑃)
coe1tmmul.u	⊢ × = (.r‘𝑅)
coe1tmmul.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)
coe1tmmul.r	⊢ (𝜑 → 𝑅 ∈ Ring)
coe1tmmul.c	⊢ (𝜑 → 𝐶 ∈ 𝐾)
coe1tmmul.d	⊢ (𝜑 → 𝐷 ∈ ℕ0)

Assertion

Ref	Expression
coe1tmmul2	⊢ (𝜑 → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))

Distinct variable groups:   𝑥, 0   𝑥,𝐶   𝑥,𝐷   𝑥,𝐾   𝑥, ↑   𝑥,𝐴   𝑥,𝑁   𝑥,𝑃   𝑥,𝑋   𝜑,𝑥   𝑥,𝑅   𝑥, ·   𝑥, ×   𝑥, ∙

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem coe1tmmul2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coe1tmmul.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
2		coe1tmmul.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		coe1tmmul.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
4		coe1tmmul.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℕ0)
5		coe1tm.k	. . . . 5 ⊢ 𝐾 = (Base‘𝑅)
6		coe1tm.p	. . . . 5 ⊢ 𝑃 = (Poly1‘𝑅)
7		coe1tm.x	. . . . 5 ⊢ 𝑋 = (var1‘𝑅)
8		coe1tm.m	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑃)
9		coe1tm.n	. . . . 5 ⊢ 𝑁 = (mulGrp‘𝑃)
10		coe1tm.e	. . . . 5 ⊢ ↑ = (.g‘𝑁)
11		coe1tmmul.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑃)
12	5, 6, 7, 8, 9, 10, 11	ply1tmcl 22591	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵)
13	1, 3, 4, 12	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵)
14		coe1tmmul.t	. . . 4 ⊢ ∙ = (.r‘𝑃)
15		coe1tmmul.u	. . . 4 ⊢ × = (.r‘𝑅)
16	6, 14, 15, 11	coe1mul 22589	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵 ∧ (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵) → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))))))
17	1, 2, 13, 16	syl3anc 1398	. 2 ⊢ (𝜑 → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))))))
18		eqeq2 2772	. . . 4 ⊢ ((((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶) ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))
19		eqeq2 2772	. . . 4 ⊢ ( 0 = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = 0 ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))
20		coe1tm.z	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
21	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑅 ∈ Ring)
22		ringmnd 20470	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
23	21, 22	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑅 ∈ Mnd)
24		ovex 7453	. . . . . . . 8 ⊢ (0...𝑥) ∈ V
25	24	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (0...𝑥) ∈ V)
26		simprr 785	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ≤ 𝑥)
27	4	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ∈ ℕ0)
28		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℕ0)
29		nn0sub 12656	. . . . . . . . . 10 ⊢ ((𝐷 ∈ ℕ0 ∧ 𝑥 ∈ ℕ0) → (𝐷 ≤ 𝑥 ↔ (𝑥 − 𝐷) ∈ ℕ0))
30	27, 28, 29	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝐷 ≤ 𝑥 ↔ (𝑥 − 𝐷) ∈ ℕ0))
31	26, 30	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − 𝐷) ∈ ℕ0)
32	27	nn0ge0d 12670	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 0 ≤ 𝐷)
33		nn0re 12615	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℝ)
34	33	ad2antrl 741	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℝ)
35	4	nn0red 12668	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℝ)
36	35	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ∈ ℝ)
37	34, 36	subge02d 11908	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (0 ≤ 𝐷 ↔ (𝑥 − 𝐷) ≤ 𝑥))
38	32, 37	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − 𝐷) ≤ 𝑥)
39		fznn0 13753	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0 → ((𝑥 − 𝐷) ∈ (0...𝑥) ↔ ((𝑥 − 𝐷) ∈ ℕ0 ∧ (𝑥 − 𝐷) ≤ 𝑥)))
40	39	ad2antrl 741	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((𝑥 − 𝐷) ∈ (0...𝑥) ↔ ((𝑥 − 𝐷) ∈ ℕ0 ∧ (𝑥 − 𝐷) ≤ 𝑥)))
41	31, 38, 40	mpbir2and 726	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − 𝐷) ∈ (0...𝑥))
42	1	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑅 ∈ Ring)
43		eqid 2760	. . . . . . . . . . . . 13 ⊢ (coe1‘𝐴) = (coe1‘𝐴)
44	43, 11, 6, 5	coe1f 22529	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐵 → (coe1‘𝐴):ℕ0⟶𝐾)
45	2, 44	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (coe1‘𝐴):ℕ0⟶𝐾)
46	45	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1‘𝐴):ℕ0⟶𝐾)
47		elfznn0 13754	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℕ0)
48	47	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℕ0)
49	46, 48	ffvelcdmd 7085	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘𝐴)‘𝑦) ∈ 𝐾)
50		eqid 2760	. . . . . . . . . . . . 13 ⊢ (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (coe1‘(𝐶 · (𝐷 ↑ 𝑋)))
51	50, 11, 6, 5	coe1f 22529	. . . . . . . . . . . 12 ⊢ ((𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵 → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾)
52	13, 51	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾)
53	52	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾)
54		fznn0sub 13690	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...𝑥) → (𝑥 − 𝑦) ∈ ℕ0)
55	54	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥 − 𝑦) ∈ ℕ0)
56	53, 55	ffvelcdmd 7085	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) ∈ 𝐾)
57	5, 15	ringcl 20477	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐴)‘𝑦) ∈ 𝐾 ∧ ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) ∈ 𝐾) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) ∈ 𝐾)
58	42, 49, 56, 57	syl3anc 1398	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) ∈ 𝐾)
59	58	fmpttd 7115	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))):(0...𝑥)⟶𝐾)
60	1	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝑅 ∈ Ring)
61	3	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝐶 ∈ 𝐾)
62	4	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝐷 ∈ ℕ0)
63		eldifi 4067	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)}) → 𝑦 ∈ (0...𝑥))
64	63, 54	syl 18	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)}) → (𝑥 − 𝑦) ∈ ℕ0)
65	64	adantl 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (𝑥 − 𝑦) ∈ ℕ0)
66		eldifsn 4747	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)}) ↔ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥 − 𝐷)))
67		simplrl 789	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℕ0)
68	67	nn0cnd 12669	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℂ)
69	47	nn0cnd 12669	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℂ)
70	69	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℂ)
71	68, 70	nncand 11674	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥 − (𝑥 − 𝑦)) = 𝑦)
72	71	eqcomd 2766	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 = (𝑥 − (𝑥 − 𝑦)))
73		oveq2 7428	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 = (𝑥 − 𝑦) → (𝑥 − 𝐷) = (𝑥 − (𝑥 − 𝑦)))
74	73	eqeq2d 2771	. . . . . . . . . . . . . . 15 ⊢ (𝐷 = (𝑥 − 𝑦) → (𝑦 = (𝑥 − 𝐷) ↔ 𝑦 = (𝑥 − (𝑥 − 𝑦))))
75	72, 74	syl5ibrcom 250	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝐷 = (𝑥 − 𝑦) → 𝑦 = (𝑥 − 𝐷)))
76	75	necon3d 2976	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑦 ≠ (𝑥 − 𝐷) → 𝐷 ≠ (𝑥 − 𝑦)))
77	76	impr 460	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥 − 𝐷))) → 𝐷 ≠ (𝑥 − 𝑦))
78	66, 77	sylan2b 606	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝐷 ≠ (𝑥 − 𝑦))
79	20, 5, 6, 7, 8, 9, 10, 60, 61, 62, 65, 78	coe1tmfv2 22594	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) = 0 )
80	79	oveq2d 7436	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = (((coe1‘𝐴)‘𝑦) × 0 ))
81	5, 15, 20	ringrz 20525	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐴)‘𝑦) ∈ 𝐾) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
82	42, 49, 81	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
83	63, 82	sylan2 605	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
84	80, 83	eqtrd 2795	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = 0 )
85	84, 25	suppss2 8217	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))) supp 0 ) ⊆ {(𝑥 − 𝐷)})
86	5, 20, 23, 25, 41, 59, 85	gsumpt 20176	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))‘(𝑥 − 𝐷)))
87		fveq2 6885	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 − 𝐷) → ((coe1‘𝐴)‘𝑦) = ((coe1‘𝐴)‘(𝑥 − 𝐷)))
88		oveq2 7428	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥 − 𝐷) → (𝑥 − 𝑦) = (𝑥 − (𝑥 − 𝐷)))
89	88	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 − 𝐷) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) = ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷))))
90	87, 89	oveq12d 7438	. . . . . . . 8 ⊢ (𝑦 = (𝑥 − 𝐷) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))))
91		eqid 2760	. . . . . . . 8 ⊢ (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))
92		ovex 7453	. . . . . . . 8 ⊢ (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))) ∈ V
93	90, 91, 92	fvmpt 6993	. . . . . . 7 ⊢ ((𝑥 − 𝐷) ∈ (0...𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))‘(𝑥 − 𝐷)) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))))
94	41, 93	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))‘(𝑥 − 𝐷)) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))))
95	28	nn0cnd 12669	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℂ)
96	27	nn0cnd 12669	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ∈ ℂ)
97	95, 96	nncand 11674	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − (𝑥 − 𝐷)) = 𝐷)
98	97	fveq2d 6889	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷))) = ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷))
99	3	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐶 ∈ 𝐾)
100	20, 5, 6, 7, 8, 9, 10	coe1tmfv1 22593	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) = 𝐶)
101	21, 99, 27, 100	syl3anc 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) = 𝐶)
102	98, 101	eqtrd 2795	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷))) = 𝐶)
103	102	oveq2d 7436	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶))
104	86, 94, 103	3eqtrd 2799	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶))
105	104	anassrs 473	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶))
106	1	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑅 ∈ Ring)
107	3	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐶 ∈ 𝐾)
108	4	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℕ0)
109	54	ad2antll 742	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) ∈ ℕ0)
110	54	nn0red 12668	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...𝑥) → (𝑥 − 𝑦) ∈ ℝ)
111	110	ad2antll 742	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) ∈ ℝ)
112	33	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑥 ∈ ℝ)
113	35	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℝ)
114	47	ad2antll 742	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℕ0)
115	114	nn0ge0d 12670	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 0 ≤ 𝑦)
116	47	nn0red 12668	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℝ)
117	116	ad2antll 742	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℝ)
118	112, 117	subge02d 11908	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (0 ≤ 𝑦 ↔ (𝑥 − 𝑦) ≤ 𝑥))
119	115, 118	mpbid 235	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) ≤ 𝑥)
120		simprl 783	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → ¬ 𝐷 ≤ 𝑥)
121	112, 113	ltnled 11457	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 < 𝐷 ↔ ¬ 𝐷 ≤ 𝑥))
122	120, 121	mpbird 260	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑥 < 𝐷)
123	111, 112, 113, 119, 122	lelttrd 11468	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) < 𝐷)
124	111, 123	gtned 11445	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐷 ≠ (𝑥 − 𝑦))
125	20, 5, 6, 7, 8, 9, 10, 106, 107, 108, 109, 124	coe1tmfv2 22594	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) = 0 )
126	125	oveq2d 7436	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = (((coe1‘𝐴)‘𝑦) × 0 ))
127	45	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (coe1‘𝐴):ℕ0⟶𝐾)
128	127, 114	ffvelcdmd 7085	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → ((coe1‘𝐴)‘𝑦) ∈ 𝐾)
129	106, 128, 81	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
130	126, 129	eqtrd 2795	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = 0 )
131	130	anassrs 473	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = 0 )
132	131	mpteq2dva 5197	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ 0 ))
133	132	oveq2d 7436	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )))
134	1, 22	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Mnd)
135	20	gsumz 19032	. . . . . . 7 ⊢ ((𝑅 ∈ Mnd ∧ (0...𝑥) ∈ V) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
136	134, 24, 135	sylancl 598	. . . . . 6 ⊢ (𝜑 → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
137	136	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
138	133, 137	eqtrd 2795	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = 0 )
139	18, 19, 105, 138	ifbothda 4520	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ))
140	139	mpteq2dva 5197	. 2 ⊢ (𝜑 → (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))))) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))
141	17, 140	eqtrd 2795	1 ⊢ (𝜑 → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450   ∖ cdif 4045  ifcif 4481  {csn 4583   class class class wbr 5102   ↦ cmpt 5185  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  ℂcc 11198  ℝcr 11199  0cc0 11200   < clt 11343   ≤ cle 11344   − cmin 11541  ℕ₀cn0 12606  ...cfz 13639  Basecbs 17387  ._rcmulr 17429   ·_𝑠 cvsca 17432  0_gc0g 17610   Σ_g cgsu 17611  Mndcmnd 18923  ._gcmg 19277  mulGrpcmgp 20360  Ringcrg 20459  var₁cv1 22494  Poly₁cpl1 22495  coe₁cco1 22496

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-ofr 7694  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8178  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-map 8849  df-pm 8850  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fsupp 9354  df-sup 9434  df-oi 9504  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-fzo 13789  df-seq 14145  df-hash 14475  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-hom 17452  df-cco 17453  df-0g 17612  df-gsum 17613  df-prds 17618  df-pws 17620  df-mre 17756  df-mrc 17757  df-acs 17759  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mhm 18978  df-submnd 18979  df-grp 19147  df-minusg 19148  df-sbg 19149  df-mulg 19278  df-subg 19333  df-ghm 19428  df-cntz 19531  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-subrng 20798  df-subrg 20822  df-lmod 21137  df-lss 21207  df-psr 22217  df-mvr 22218  df-mpl 22219  df-opsr 22221  df-psr1 22498  df-vr1 22499  df-ply1 22500  df-coe1 22501

This theorem is used by:  coe1tmmul2fv  22597  coe1sclmul2  22603